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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Einstein manifold ： ウィキペディア英語版 Einstein manifold In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field equations (with cosmological constant), although the dimension, as well as the signature, of the metric can be arbitrary, unlike the four-dimensional Lorentzian manifolds usually studied in general relativity. If ''M'' is the underlying ''n''-dimensional manifold and ''g'' is its metric tensor the Einstein condition means that :$\backslash mathrm\; =\; k\backslash ,g,$ for some constant ''k'', where Ric denotes the Ricci tensor of ''g''. Einstein manifolds with ''k'' = 0 are called Ricci-flat manifolds. ==The Einstein condition and Einstein's equation== In local coordinates the condition that (''M'', ''g'') be an Einstein manifold is simply :$R\_\; =\; k\backslash ,g\_.$ Taking the trace of both sides reveals that the constant of proportionality ''k'' for Einstein manifolds is related to the scalar curvature ''R'' by :$R\; =\; nk\backslash ,$ where ''n'' is the dimension of ''M''. In general relativity, Einstein's equation with a cosmological constant Λ is :$R\_\; -\; \backslash fracg\_R\; +\; g\_\backslash Lambda\; =\; 8\backslash pi\; T\_,$ written in geometrized units with ''G'' = ''c'' = 1. The stress–energy tensor ''T''''ab'' gives the matter and energy content of the underlying spacetime. In a vacuum (a region of spacetime with no matter) ''T''''ab'' = 0, and one can rewrite Einstein's equation in the form (assuming ''n'' > 2): :$R\_\; =\; \backslash frac\backslash ,g\_.$ Therefore, vacuum solutions of Einstein's equation are (Lorentzian) Einstein manifolds with ''k'' proportional to the cosmological constant. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Einstein manifold」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.